Mind, Brain, & Education by Solution Tree

Sousa

David A. Sousa, EdD, is an international consultant in educational neuroscience and author of a dozen books that suggest ways that educators and parents can translate current brain research into strategies for improving learning. A member of the Cognitive Neuroscience Society, he has conducted workshops in hundreds of school districts on brain research, instructional skills, and science education at the preK–12 and university levels. He has made presentations to more than 100,000 educators at national conventions of educational organizations and to regional and local school districts across the United States, Canada, Europe, Australia, New Zealand, and Asia. Dr. Sousa’s popular books for educators include: How the Brain Learns, third edition; How the Special Needs Brain Learns, second edition; How the Gifted Brain Learns; How the Brain Learns to Read; How the Brain Influences Behavior; and How the Brain Learns Mathematics, which was selected by the Independent Publishers Association as one of the best professional development books of 2008.

—From the introduction by David A. Sousa

Mind, Brain, and Education brings together the visionaries in educational neuroscience, an emerging field that unites psychology, neuroscience, and pedagogy. The contributors explain the significant research on how the brain develops and learns, explore its implications for educational practice, and offer new ways of thinking about intelligence and academic ability. David A. Sousa explains the origins of educational neuroscience and the impact of brain research on education. Michael I. Posner describes how advances in neuroimaging technology led to deeper understandings of the brain. Judy Willis shares practical classroom strategies for applying what is known about the brain. Mary Helen Immordino-Yang and Matthias Faeth present a neuroscientific view of how emotions affect learning and suggest practices to improve the emotional and cognitive aspects of classroom learning. Diane L. Williams reviews the neuroscience research and debunks popular myths about learning language. John Gabrieli, Joanna A. Christodoulou, Tricia O’Loughlin, and Marianna D. Eddy examine what we know about how the brain learns to read, why some children struggle to read, and what research tells us about reading interventions. Donna Coch explains what neuroscience has revealed about the interaction of the visual and auditory processing systems, the development of the alphabetic principle, semantics, and comprehension. Keith Devlin explores what we know to date about innate number sense and proposes instructional approaches

in mathematics based on recent neuroscience research. Stanislas Dehaene discusses the three networks used to evaluate the number of a set of objects, how humans innately approximate number, and how this knowledge can be used to help students learn mathematics. Daniel Ansari describes how concrete neurological differences contribute to individuals’ varying mathematical ability. Mariale M. Hardiman presents neuroscience research about the nature of creativity and how it can be cultivated through the arts, then introduces a framework for integrating creativity across content and grade levels.

Sousa EDITOR

6 The Leading Edge is the undefined space where leaders venture to impact change—it is the place where transformation begins. The Leading Edge™ series unites education authorities from around the globe and asks them to confront the important issues that affect teachers and administrators—the important issues that profoundly impact student success.

mind, brain, & education Neuroscience Implications for the Classroom

Joanna A. Christodoulou Donna Coch Stanislas Dehaene Keith Devlin

Kurt W. Fischer and Katie Heikkinen argue that new knowledge about the brain necessitates collaboration between neuroscientists and educators to shift current thinking about teaching and learning.

solution-tree.com

David A.

Daniel Ansari

Marianna D. Eddy Matthias Faeth Kurt W. Fischer John Gabrieli

Sousa

Dr. Sousa is past president of the National Staff Development Council. He has received numerous awards from professional associations, school districts, and educational foundations for his commitment to research, staff development, and science education. He recently received the Distinguished Alumni Award and an honorary doctorate from Massachusetts State College (Bridgewater), and an honorary doctorate from Gratz College in Philadelphia.

“The brain remains an enormously complex wonder that still guards many secrets. But we are slowly pulling back the veil and gaining insights that have implications for teaching and learning. . . . Exploring these chapters will give the reader a sense of where this new field of educational neuroscience is now, and where it is headed.”

mind, brain, & education

Neuroscience Implications for the Classroom

David A. Sousa

EDITOR

Neuroscience Implications for the Classroom

mind, brain, & education

David A.

Mariale M. Hardiman Katie Heikkinen Mary Helen Immordino-Yang Tricia O’Loughlin Michael I. Posner David A. Sousa Diane L. Williams Judy Willis

The experts contributing to this anthology do not prescribe one method to transact change. They embrace the mission, trusting that teachers and administrators—the true change leaders—will venture to the Leading Edge to embrace challenges and opportunities that will guarantee the success of their students. You are invited to the Leading Edge to explore Mind, Brain, and Education: Neuroscience Implications for the Classroom. This sixth book in the series examines neuroscience’s growing knowledge of how the brain develops and functions, and explores the field’s implications for pedagogy and the classroom. The contributors investigate such questions as: What are the neurological foundations of learning and of individual differences in learning? How did educators get involved with neuroscience, and where might this involvement lead? What does neuroscience reveal about the brain’s ability to learn and use written and spoken language, to learn and use mathematics, and to think creatively? How can educational neuroscience improve the teaching of these abilities? The Leading Edge series also includes On Common Ground: The Power of Professional Learning Communities; Ahead of the Curve: The Power of Assessment to Transform Teaching and Learning; Change Wars; On Excellence in Teaching; and 21st Century Skills: Rethinking How Students Learn.

Copyright ÂŠ 2010 by Solution Tree Press All rights reserved, including the right of reproduction of this book in whole or in part in any form. 555 North Morton Street Bloomington, IN 47404 800.733.6786 (toll free) / 812.336.7700 FAX: 812.336.7790 email: info@solution-tree.com solution-tree.com Printed in the United States of America 14 13 12 11 10

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Library of Congress Cataloging-in-Publication Data Mind, brain, and education : neuroscience implications for the classroom / David A. Sousa ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-1-935249-63-4 1. Learning, Psychology of. 2. Cognitive learning. 3. Teaching--Psychological aspects. 4. Brain. I. Sousa, David A. LB1060.M5587 2010 370.15â&#x20AC;&#x2122;23--dc22 2010016078 Solution Tree Jeffrey C. Jones, CEO & President Solution Tree Press President: Douglas M. Rife Publisher: Robert D. Clouse Vice President of Production: Gretchen Knapp Managing Production Editor: Caroline Wise Proofreader: Elisabeth Abrams Text Designer: Amy Shock Cover Designer: Pamela Rude

Table of Contents About the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Introduction David A. Sousa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1

How Science Met Pedagogy David A. Sousa . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 2

Neuroimaging Tools and the Evolution of Educational Neuroscience Michael I. Posner . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 3

The Current Impact of Neuroscience on Teaching and Learning Judy Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 4

The Role of Emotion and Skilled Intuition in Learning Mary Helen Immordino-Yang and Matthias Faeth . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 5

The Speaking Brain Diane L. Williams . . . . . . . . . . . . . . . . . . . . . . . . 85

Chapter 6

The Reading Brain John Gabrieli, Joanna A. Christodoulou, Tricia Oâ&#x20AC;&#x2122; Loughlin, and Marianna D. Eddy . . 113

Chapter 7

Constructing a Reading Brain Donna Coch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Chapter 8

The Mathematical Brain Keith Devlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Chapter 9

The Calculating Brain Stanislas Dehaene . . . . . . . . . . . . . . . . . . . . . . . 179

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Chapter 10

The Computing Brain Daniel Ansari . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Chapter 11

The Creative-Artistic Brain Mariale M. Hardiman . . . . . . . . . . . . . . . . . . . 227

Chapter 12

The Future of Educational Neuroscience Kurt W. Fischer and Katie Heikkinen . . . . . . . 249

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

About the Editor

David A. Sousa David A. Sousa, EdD, is an international consultant in educational neuroscience and author of a dozen books that suggest ways that educators and parents can translate current brain research into strategies for improving learning. A member of the Cognitive Neuroscience Society, he has conducted workshops in hundreds of school districts on brain research, instructional skills, and science education at the preK–12 and university levels. He has made presentations to more than 100,000 educators at national conventions of educational organizations and to regional and local school districts across the U.S., Canada, Europe, Australia, New Zealand, and Asia. Dr. Sousa has a bachelor’s degree in chemistry from Massachusetts State College at Bridgewater, a Master of Arts in Teaching degree in science from Harvard University, and a doctorate from Rutgers University. His teaching experience covers all levels. He has taught senior high school science and served as a K–12 director of science, a supervisor of instruction, and a district superintendent in New Jersey schools. He has been an adjunct professor of education at Seton Hall University and a visiting lecturer at Rutgers University. Prior to his career in New Jersey, Dr. Sousa taught at the American School of Paris and served for five years as a Foreign Service Officer and science advisor at the USA diplomatic missions in Geneva and Vienna. v

mind, brain, & education

Dr. Sousa has edited science books and published dozens of articles in leading journals on staff development, science education, and educational research. His popular books for educators include: How the Brain Learns, third edition; How the Special Needs Brain Learns, second edition; How the Gifted Brain Learns; How the Brain Learns to Read; How the Brain Influences Behavior; and How the Brain Learns Mathematics, which was selected by the Independent Publishers Association as one of the best professional development books of 2008. The Leadership Brain suggests ways for educators to lead todayâ&#x20AC;&#x2122;s schools more effectively. His books have been published in French, Spanish, Chinese, Arabic, and several other languages. Dr. Sousa is past president of the National Staff Development Council. He has received numerous awards from professional associations, school districts, and educational foundations for his commitment to research, staff development, and science education. He recently received the Distinguished Alumni Award and an honorary doctorate from Massachusetts State College (Bridgewater), and an honorary doctorate from Gratz College in Philadelphia. Dr. Sousa has been interviewed by Matt Lauer on the NBC TODAY show and by National Public Radio about his work with schools using brain research. He makes his home in south Florida.

Introduction

David A. Sousa You hold in your hands a historical publication. This book is the first to bring together some of the most influential scholars responsible for giving birth to a new body of knowledge: educational neuroscience. This newbornâ&#x20AC;&#x2122;s gestation period was not easy. Lasting for several decades, it was difficult and often contentious. Identifying the parents was elusive at best, as more than a few prominent candidates denied kinship. Just naming the offspring was a daunting challenge and more exhausting than herding cats. Nevertheless, the birth occurred recently with the help of the visionaries who have contributed to this book. And teaching will never be the same again. For centuries, the practice of medicine The birth of educational was an art form, driven by creativity and hope, neuroscience occurred with but with little understanding of how to cure the help of the visionaries disease. Physicians tried certain treatments who have contributed to and administered specific herbs or potions this book. And teaching will based largely on their previous experiences or never be the same again. on advice from colleagues. They did not know why some treatments worked on one individual and not another, or why they worked at all. Their practice was essentially trial and error, with an occasional stroke of luck. All that changed when Alexander Fleming discovered penicillin in 1928. Although it took more than a decade for penicillin to be mass produced, it gave physicians their first drug for fighting several serious diseases. Furthermore, by understanding how penicillin disrupted the reproduction of bacteria, physicians could make informed decisions about treatment. Medical practice was not just an art form, but had also crossed the threshold into the realm of a science. 1

mind, brain, & education

Today, a similar story can also apply to teaching. Teachers have taught for centuries without knowing much, if anything, about how the brain works. That was mainly because there was little scientific understanding or credible evidence about the biology of the brain. Teaching, like early medicine, was essentially an art form. Now, thanks to the development of imaging techniques that look at the living brain at work, we have a better understanding of its mechanisms and networks. Sure, the brain remains an enormously complex wonder that still guards many secrets. But we are slowly pulling back the veil and gaining insights that have implications for teaching and learning. Since the 1990s, educators all over the world have come to recognize that there is a rapidly increasing knowledge base about the human brain. Through numerous articles, books, videos, and other presentations, they have also become aware that some of this knowledge could inform educational practice. What many educators may not realize, however, is that researchers and practicing educators have worked diligently to establish a legitimate scientific area of study that overlaps psychology, neuroscience, and pedagogy. The result is educational neuroscience (see fig. I.1).

Until recently, teaching, like early medicine, was essentially an art form.

Psychology The study of mental processes responsible for cognition and behavior (mind)

Educational Neuroscience

Pedagogy The study of the art and science of teaching (education)

Neuroscience The study of the brainâ&#x20AC;&#x2122;s development, structure, and function (brain)

Figure I.1: The emergence of educational neuroscience at the Âintersection of psychology, neuroscience, and pedagogy.

Introduction

As exciting as all this seems, understandable skepticism and numerous questions still exist. What specific research applies to pedagogy? Will it benefit students? How do we know the research is being interpreted accurately? Can it be reasonably adapted to our schools and classrooms? The emergence of a new body of knowledge should be cause for celebration and, in this case, especially among educators. Here in these pages you will discover why we are celebrating. Some of the major pioneers in educational neuroscience explain recent discoveries about the human brain and discuss the influence these discoveries can have on teaching and learning—some now, some in the near future. The contributors will explore questions such as these: • How and when did educators get involved with neuroscience? • How does neuroimaging contribute to our understanding of how the brain learns? • In what ways is neuroscience research already having an impact on teaching and learning? • In what ways do emotions affect our ability to learn? • How does a child acquire spoken language? • What brain networks are required to learn how to read? • If we are born with an innate number sense, how can teachers use this to help students learn arithmetic and mathematics more successfully? • How does the brain represent quantity and numbers? • What is creativity, and can it be taught? • In what ways do the arts contribute to brain development? • What does the future hold for educational neuroscience? Writing about all the areas researchers are currently investigating that could have an impact on educational neuroscience would fill a volume ten times the size of this one. So it became necessary to focus on those areas that have the greatest potential for affecting educational practice now or in the near future.

mind, brain, & education

We begin with some background on how this new area of study evolved over the past few decades. In chapter 1, I discuss how and why I and a few other educators got so involved following the explosion of brain research in the 1980s and 1990s. Our entry into this new area of study generated considerable controversy, but our tenacity paid off. The main reason for that explosion was the development of imaging technology that allowed researchers to peek inside the workings of the living brain. Michael Posner was one of the pioneers in using the new imaging devices, and he explains their contribution to neuroscientific research in chapter 2. Some of the research findings from neuroscience are already being used in educational practice. In chapter 3, Judy Willis, a medical doctor turned classroom teacher, writes about those areas that are already having an impact on instruction and offers numerous suggestions for teachers to consider. Students do not just develop intellectually in schools, but also socially and emotionally. Despite our understanding that emotions have an impact on learning, some teachers are still unsure how to incorporate emotions into their lessons. Mary Helen ImmordinoYang and Matthias Faeth cite five major contributions in chapter 4 that neuroscience has made to the research on how emotions affect learning, and they suggest three strategies that have proven effective. Speaking and learning to read are among the early skills that young children learn. In chapter 5, Diane Williams explains what neuroscience research has revealed about the cerebral networks involved in learning spoken language. She debunks popular myths about learning language and discusses some major implications that this research has for teaching and learning. Because reading is one of the most challenging tasks the young brain will undertake, it has gained a lot of attention from neuroscientists and cognitive psychologists. Consequently, we have devoted two chapters to this topic. In chapter 6, John Gabrieli and his colleagues review the major research findings and current understandings about how a childâ&#x20AC;&#x2122;s brain learns to read, what is different in the brain of a child who struggles to read, and how the neuroscience of reading may come to play an important role in education. Donna Coch in chapter

Introduction

7 also discusses the complex processes involved as the young brain learns to read, but focuses more on the role of the visual and auditory processing systems as well as the development of the alphabetic principle, semantics, and comprehension. Another area of great interest to neuroscience researchers is how the brain represents quantity and how it engages neural networks to learn to carry out arithmetic and mathematical computations. Three renowned researchers in this area offer insights for educators. Keith Devlin suggests in chapter 8 that the brainâ&#x20AC;&#x2122;s strength as a pattern-seeker accounts for many of the difficulties people have with basic arithmetic operations. He offers some proposals to educators on instructional approaches in mathematics based on the recent neuroscience research. In chapter 9, Stanislas Dehaene explains how neuroimaging has helped us to understand the three networks our brain uses to evaluate the number of a set of objects and suggests ways this and other discoveries can be used to help students learn arithmetic and mathematics. Not all brains do well with mathematics, however; in chapter 10, Daniel Ansari reviews what is currently known about how the brain computes. He discusses how the brains of individuals with and without mathematical difficulties differ both functionally and structurally. In addition, he suggests ways in which research findings may inform both the thinking and practice of educators. What is creativity, and can it be taught? How do the arts help students develop competency in other subject areas? These questions are of particular importance because in too many school districts, arts are still thought of as frill subjects and are thus easy targets when budgets get tight. In chapter 11, Mariale Hardiman addresses these important questions and offers suggestions that teachers can use to incorporate the arts in all subjects and at all grade levels. Because many of the authors refer to specific regions of the brain in their discussion, we have included two diagrams (figs. I.2 and I.3, pages 6â&#x20AC;&#x201C;7) that should help readers locate these regions. In addition, we have included a glossary at the end of the book (pages 271â&#x20AC;&#x201C;274) that defines the less-familiar scientific terms used by the authors. With all these promising research findings, where do we go from here? To answer that question, in chapter 12, Kurt Fischer and Katie

mind, brain, & education

Heikkinen suggest that new ways of thinking about teaching will need to emerge if educational neuroscience is to meet its promise in the future. Exploring these chapters will give the reader a sense of where this new field of educational neuroscience is now, and where it is headed. These authors have been instrumental in supporting this emerging area of study. Their ideas and continuing research are sure to help educators find applications to their practice that will benefit all students. Parietal lobe

Frontal lobe

Angular gyrus Occipital lobe Temporal lobe

Brocaâ&#x20AC;&#x2122;s area

Visual word form area Cerebellum

Wernickeâ&#x20AC;&#x2122;s area

Brain stem

Figure I.2: The exterior regions of the human brain referred to in this book.

Introduction Thalamus Corpus callosum

Frontal lobe Orbitofrontal cortex Amygdala Brain stem Hippocampus

Figure I.3: A cross-section of the human brain showing the major regions referred to in this book.

Keith Devlin Keith Devlin, PhD, is a cofounder and Executive Director of Stanford University’s H-STAR institute, a cofounder of the Stanford Media X research network, and Senior Researcher at the university’s Center for the Study of Language and Information. He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Dr. Devlin’s other research interests include theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written twenty-eight books and over eighty published research articles. He is the recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. In 2003, he was recognized by the California State Assembly for his “innovative work and longtime service in the field of mathematics and its relation to logic and linguistics.” He is “the Math Guy” on National Public Radio. In this chapter, Dr. Devlin discusses what we know to date about how the young brain uses innate number sense to learn to process numbers and eventually become efficient at arithmetic operations. He suggests that the brain’s strength as a pattern-seeker accounts for many of the difficulties people have with basic arithmetic operations. The mental processes required for algebra and beyond are also examined, and he offers some proposals to educators on instructional approaches in mathematics based on recent neuroscience research.

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The Mathematical Brain Keith Devlin Mathematics teachers—at all education levels—face two significant obstacles: 1. We know almost nothing about how people do mathematics. 2. We know almost nothing about how people learn how to do mathematics. Even professional mathematicians, who spend most waking hours of every day thinking about mathematics, are not aware of what is going on in their minds when they solve a problem. One minute the problem seems intractable, the next everything has miraculously fallen into place and the answer seems obvious. Never again will they truly understand why they had been unable to see it earlier. It follows that mathematics teaching remains largely an art—with good teachers relying heavily on tradition, instinct, experience, and “received wisdom” gained from other teachers—rather than a practice based on established scientific theory and evidence. This situation is somewhat reminiscent of the state of medicine in the nineteenth century. It remains to be seen whether our understanding of the human brain and the human mind will develop to a point that the field of education reaches a level comparable to today’s medical practice. Cognitive neuroscience and cognitive psychology, the equivalents of the chemistry, biology, and physiology on which modern medicine is based, are still relatively young disciplines. In 163

Chapter 8

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What We Know The greatest advances in our understanding of mathematical ability have undoubtedly been in the acquisition of the whole number concept and its arithmetic. Starting with Karen Wynn’s breakthrough discoveries at MIT in the early 1990s (Wynn, 1992) and continuing with work of Stanislas Dehaene (Dehaene, 1997), Brian Butterworth (Butterworth, 1999), and others, we have learned that: • The brain of humans and some other species has a “number sense,” a seemingly analog sense of relative sizes of collections. • The brain of humans and some other species has a capacity to form a concept of (discrete) whole number. • The brain of humans and some other species can distinguish a correct answer from an incorrect one when presented with a scenario involving arithmetic of small whole numbers (1, 2, and 3). • Numbers and arithmetic beyond 3 require the use of language. The greatest advances in our understanding of mathematical ability have been in the acquisition of whole number concept and its arithmetic.

Studies using fMRI indicate that these number capacities appear to be localized in (or at least depend on) particular brain areas. Moreover, damage to certain regions of the brain can impair or destroy any sense of number, leaving other capacities intact, such as language or logical reasoning.

Number Sense The number sense exhibits what is often referred to as a logarithmic nature, in that the speed and accuracy with which a subject

the meantime, we have to rely on hypotheses and speculative theories based on what little we do know about how the brain works—or more precisely, how the mind works, because our knowledge of brain function is actually fairly significant. However, it is important not to confuse hypotheses and speculative theories with guesses. We may not yet have firm conclusions based on hard evidence, but we know a lot more than we did just a few decades ago.

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Small Numbers Human babies as young as two days after birth exhibit an innate knowledge of the basic arithmetic facts: 1 + 1 = 2, 1 + 2 = 3, 1 - 1 = 0, 2 - 1 = 1, 3 - 1 = 2, 3 - 2 = 1, 2 - 2 = 0, 3 - 3 = 0. The general strategy for testing this capacity is, first, to present the subject with a visual depiction of objects being added to and taken away from a collection in such a way that the final outcome is initially hidden from the subject’s view and, second, to measure the surprise the subject exhibits when the outcome is revealed. (Hiding the final step allows for manipulation of the result unseen by the subject.) The surprise response can be measured Human babies as young as two days after birth exhibit by eye dilation, facial expression, and body an innate knowledge of reaction captured on video, or by the sucking basic arithmetic facts. reflex on a nipple attached by a transponder to a computer. A child subject will exhibit greater surprise when presented with a wrong answer, such as 1 + 2 = 2, as opposed to the correct answer of 1 + 2 = 3. In the first series of such experiments, Karen Wynn (1992) showed her young subjects puppets being put onto and taken from a small puppet theater. The subject first saw the stage empty and then a screen was raised in front of the stage, behind which puppets were visibly introduced and removed. Variations of this procedure eliminated any possibility that the subject’s reactions were based on anything other than number.

Beyond 3 The innate precision shown for addition and subtraction for whole numbers up to and including 3 does not extend to greater numbers. Arithmetic ability beyond 3 seems to involve language (at least the

can distinguish the larger of two collections (or two numerals in the case of human adults) diminishes as the size of the two collections increases relative to their difference. For example, selecting the larger of two collections of five and eight objects is faster and more accurate than for two collections of nineteen and twenty-two, even though the difference in each case is three.

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ability to assign symbolic names to objects and to carry out binary symbolic reasoning) and requires training.

1. Children being taught mathematics in a language other than the one in which they first learned the basic number facts (in particular, number names and the multiplication tables) are likely to be at a disadvantage. 2. Asian children enjoy a considerable advantage over their counterparts growing up in the West. The basic number words in Chinese, Japanese, and Korean are a single syllable and combine in a way parallel to the symbolic rules of the decimal number system. For example, 21 would be expressed as “two-ten-one.” In English, French, Spanish, and other Western languages, the naming system is more complex, with the need to master special words each time a multiple of ten is reached—eleven, twelve, thirteen, twenty, thirty, and so on—and in some languages special constructions, such as “quatre vingt trois” (literally “four-twenty and three”) for 83 in French (see Dehaene, 1997).

Implications for Teaching To summarize, the picture of arithmetic learning that seems to emerge from the research is that: 1. Humans are born with an innate number sense, accurate in the range 1 to 3, increasingly less accurate and of a more analog nature beyond three. 2. With training, humans can make use of their language capacity to handle with precision numbers beyond 3—indeed, well beyond. Mathematics educators sometimes disagree as to the most effective way to provide the training referred to in the second item. The disagreement arises from the fact that learning about numbers

In particular, mental storage of numbers and access to them appears to be by way of the symbolic names used during the initial learning process. If so, this has two significant implications for learning arithmetic:

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and how to do arithmetic involves both bottom-up and top-down components. Both of these are required and must mesh in order for successful learning to take place.

The familiar system we use to write numbers, using the ten digits 0, 1, 2, . . . , 9, and the algorithms we use to perform basic arithmetic calculations with numbers so written, is known as the Hindu-Arabic decimal number system. It was developed in India during the first six or seven centuries CE and refined by Arabic-speaking scholars before finding its way into Western Europe in the 13th century. It is neither innate nor naturally occurring; it is a human invention and is built on several essentially arbitrary conventions. It became the dominant system, replacing several others, because of its simplicity and efficiency. Arguably the greatest drawback of the The greatest drawback to Hindu-Arabic system is that it takes a consid- our system of numbers is erable effort to learn it before you can use it that it takes considerable efficiently. First, you have to learn how to read effort to learn it before and write numbers in the decimal place-value you can use it efficiently. system. Then you have to commit to memory several basic number facts, most notably the multiplication tables. Then you have to learn the algorithms used to compute. Some of the controversy surrounding mathematics education involves the need for rote learning. At one extreme are those who argue that mathematics education should focus entirely on conceptual

First, we have an innate number sense on which we build our formal number concept (bottom-up), and yet we have to use language to impose precision and generate numbers (top-down). Second, the basic operations of arithmetic—addition, subtraction, multiplication, and division—correspond to things we witness and experience in the world and to activities we do in the world (such as combining collections of objects together, scaling, sharing out, and so forth). Thus, arithmetic can be seen to build on our experience of the world—a bottom-up process. But in order to perform the calculations required to reach precise answers about those activities in the world, we have to use symbolic procedures (algorithms) invented by humans—a top-down process.

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Neuroscience has shown us that learning involves the creation or strengthening of various neural pathways in the brain, and that this is achieved by repetition. In fact, as far as we know, repetition is the only way the brain can learn (Deacon, 1997). This understanding provides strong support for those who argue that basic arithmetic education must include rote learning. Without rote learning, no one would ever learn the multiplication tables. On the other hand, numerous studies have shown that rote learning alone produces a narrow and brittle form of knowledge, whereby the individual can reproduce—or recite—what has been learned (and thus can pass the test) but does not necessarily understand the new information and is unable to make practical use of it. To produce applicable knowledge and generate useful skills, rote learning must be accompanied by understanding. Note: accompanied by understanding, not replaced by it. Mathematics is an activity—you do it. Like driving a car or skiing, it depends on mastery of basic skills. The necessary understanding needs to be in addition to, indeed coupled with, rote learning. Neither can succeed without the other.

To produce applicable knowledge and generate useful skills, rote learning must be accompanied by understanding.

Learning Basic Number Facts Because of the way the brain works and the way it creates and handles numbers through language, learning the multiplication tables is essentially a linguistic task, not a mathematical one. Even today, more than fifty-five years after I “learned my tables,” I still recall the product of any two single-digit numbers by reciting that entry in the table in my head. I remember the sound of the number words spoken,

understanding, and that there is no place for rote learning. At the other extreme are those who say that mathematics education should concentrate on rote learning the basic number facts and practice repeatedly the procedures—an approach often referred to by proponents as “drill and skill” and by opponents as “drill and kill.” The evidence suggests that the best (and in my view the only viable) approach lies somewhere in between these extremes.

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not the numbers themselves. Indeed, I believe the pattern I hear in my head is precisely the one I learned when I was seven years old!

Why do we have such difficulty? Discounting the times tables for 1 and 10 as presenting no difficulty, the entire collection of multiplication tables amounts to only sixty-four separate facts (each one of 2, 3, 4, . . . , 9 multiplied by each one of 2, 3, 4, . . . , 9). Most people have little problem with the times tables for 2 or 5. Further discounting those tables leaves only thirty-six single-digit multiplications that take some effort to memorize. In fact, because you can swap the order in multiplication (4 × 7 is the same as 7 × 4), the real total is only half of that, or eighteen. To put that figure of eighteen simple facts in perspective, consider that by the age of six, a typical American child will have learned to use and recognize between 13,000 and 15,000 words. An American adult has a comprehension vocabulary of about 100,000 words and makes active and fluent use of 10,000 to 15,000. Then there are all the other things we remember: people’s names, phone numbers, addresses, book titles, movie titles, and so forth. Moreover, we learn these facts with hardly any difficulty. We certainly don’t have to recite words and their meanings over and over the way we do our multiplication tables. In short, most of the time there is nothing wrong with our memories. Not so when it comes to the eighteen key facts of multiplication. Why?

The Brain as Pattern-Seeker The human mind is a pattern recognizer. Human memory works by association—one thought leads to another. The ability to see patterns and similarities is one of the greatest strengths of the human mind. This is very different from the way a digital computer works. Computers are good at precise storage and retrieval of information and exact calculation. A modern computer can perform billions of

Understanding how we handle numbers can explain why, despite many hours of practice, most people encounter great difficulty with the multiplication tables. Ordinary adults of average intelligence make mistakes roughly 10 percent of the time. Some multiplications, such as 8 × 7 or 9 × 7, can take up to two seconds, and the error rate goes up to 25 percent. (Common wrong answers are 8 × 7 = 54 and 9 × 7 = 64.)

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The reason we have such trouble is that we remember the tables linguistically, and as a result, many of the different entries interfere with one another. A computer is so dumb that it treats 7 × 8 = 56, 6 × 9 = 54, and 8 × 8 = 64 as distinct from each other. But the human mind sees similarities between these three multiplications, particularly linguistic similarities in the rhythm of the words when we recite them aloud. Our difficulty in trying to keep these three equations separate does not indicate a weakness of our memory but of one of its major strengths—its ability to see similarities. When we see the pattern 7 × 8, it activates several other patterns, among which are likely to be 48, 56, 54, 45, and 64. Stanislas Dehaene makes this point brilliantly in The Number Sense (Dehaene, 1997) with the following example. Suppose you had to remember the following three names and addresses: • Charlie David lives on Albert Bruno Avenue. • Charlie George lives on Bruno Albert Avenue. • George Ernie lives on Charlie Ernie Avenue. Remembering just these three facts looks like quite a challenge. There are too many similarities, and as a result, each entry interferes with the others. But these are just entries from the multiplication tables. Let the names Albert, Bruno, Charlie, David, Ernie, Fred, and George stand for the digits 1, 2, 3, 4, 5, 6, and 7, respectively, and replace the phrase “lives on” by the equals sign, and you get these multiplications: • 3 × 4 = 12 • 3 × 7 = 21 • 7 × 5 = 35

multiplications in a single second, getting each one right. But despite an enormous investment in money, talent, and time over fifty years, attempts to develop computers that can recognize faces or indeed make much sense at all of a visual scene have largely failed. Humans, on the other hand, recognize faces and scenes with ease, because human memory works by pattern association. For the same reason, however, we can’t do some things that computers do with ease, including remembering multiplication tables.

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It’s the pattern interference that causes our problems.

Linguistic pattern similarities also interfere with retrieval from the multiplication table when we are asked for 5 × 6 and answer 36 or 56. Somehow, reading the 5 and the 6 brings to mind both incorrect answers. People do not make errors such as 2 × 3 = 23 or 3 × 7 = 37. Because the numbers 23 and 37 do not appear in any multiplication table, our associative memory does not bring them up in the context of multiplication. But 36 and 56 are both in the table, so when our brain sees 5 × 6, both numbers are activated. In other words, much of our difficulty with multiplication comes from two of the most powerful and useful features of the human mind: pattern recognition and associative memory. To put it another way, millions of years Much of our difficulty of evolution have equipped us with a brain with multiplication comes that has particular survival skills. Part of that from two of the most endowment is that our minds are very good at powerful and useful recognizing patterns, seeing connections, and features of the human making rapid judgments and inferences. All of mind: pattern recognition and associative memory. these modes of thinking are essentially “fuzzy.” Our brains are not at all suited to the kinds of precise manipulations of information that arise in arithmetic—they did not evolve to do arithmetic. To do arithmetic, we have to marshal mental circuits that developed (or were selected during evolution) for quite different reasons. It’s like using the edge of a small coin to turn a screw. Sure, you can do it, but it’s slow, and the outcome is not always perfect.

Pattern interference is also the reason why it takes longer to realize that 2 × 3 = 5 is false than to realize that 2 × 3 = 7 is wrong. The former equation is correct for addition (2 + 3 = 5), and so the pattern “2 and 3 make 5” is familiar to us. There is no familiar pattern of the form “2 and 3 make 7.” We see this kind of pattern interference in the learning process of young children. By the age of seven, most children know by heart many additions of two digits. But as they start to learn their multiplication tables, the time it takes them to answer a single-digit addition sum increases, and they start to make errors, such as 2 + 3 = 6.

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How Our Brains Store and Access Numbers

The hypothesis they set out to establish was this: that arithmetical tasks that require an exact answer depend on our linguistic faculty. In particular, such tasks use the verbal representations of numbers, whereas tasks that involve estimation or require an approximate answer do not make use of the language faculty. To test this hypothesis, the researchers assembled a group of English-Russian bilinguals and taught them some new two-digit addition facts in one of the two languages. The subjects were then tested in one of the two languages. For questions that required an exact answer, when both the instruction and the question were in the same language, subjects answered in 2.5 to 4.5 seconds but took a full second longer when the languages were different. The experimenters concluded that the subjects used the extra second to translate the question into the language in which the facts had been learned. However, when the question asked for an approximate answer, the language of questioning did not affect the response time. The researchers also monitored the subjects’ brain activity throughout the testing process. When the subjects were answering questions that asked for approximate answers, the greatest brain activity was in the two parietal lobes—the regions that house number sense and support spatial reasoning. Questions requiring an exact answer, however, elicited far more activity in the frontal lobe, where speech is controlled.

We learn the multiplication tables by using our ability to remember patterns of sound. So great is the effort required to learn the tables (because of the interference effects) that people who learn a second language generally continue to do arithmetic in their first language. No matter how fluent they become in their second language (and many people reach the stage of thinking entirely in whichever language they are conversing in), it’s easier to slip back into their first language to calculate and then translate the result back to the second language than to try to relearn the multiplication table in the second language. This idea formed the basis of an ingenious experiment that Dehaene and his colleagues performed in 1999 to confirm that we use our language faculty to do arithmetic.

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Algebra

Algebraic thinking is not just arithmetic with letters standing for numbers. It is a different kind of thinking. Algebra is thinking logically about numbers rather than computing with numbers. Those x’s and y’s usually denote numbers in general, not particular numbers. In algebra, we use Algebra is thinking logically about numbers; arithmetic analytic, qualitative reasoning about numbers, is computing with numbers. whereas in arithmetic we use numerical, quantitative reasoning with numbers. For example, you need to use algebra if you want to write a macro to calculate the cells in a spreadsheet such as with Microsoft Excel. It doesn’t matter whether the spreadsheet is for calculating scores in a sporting competition, keeping track of your finances, running a business, or figuring out the best way to equip your character in World of Warcraft. You need to think algebraically, rather than arithmetically, to set up the spreadsheet to do what you want. When students start to learn algebra, they inevitably try to solve problems by thinking arithmetically. That’s a natural thing to do, given all the effort they have put into mastering arithmetic. And at first, when the algebra problems they meet are particularly simple (that’s the teacher’s classification), this approach works. In fact, the stronger a student is at arithmetic, the further he or she can progress in algebra using arithmetical thinking. (Many students can solve the quadratic equation x2 = 2x + 15 using basic arithmetic rather than algebra.) Paradoxically, or so it may seem, those better students may thus find it harder to learn algebra. To do algebra, for all but the most basic examples, students must stop thinking arithmetically and learn to think algebraically.

Beginning with algebra, most mathematics beyond arithmetic presents the human mind with an additional challenge: abstraction. The basic building blocks of arithmetic, numbers, are abstractions, but abstractions that are tied closely to concrete things in the world in which we live. We count things, measure things, buy things, make things, use the telephone, go to the bank, check the baseball scores, and so on. With algebra, however, the level of abstraction is greater.

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This does not mean that teachers should not try to provide rationales and explanations for those rules. But the real understanding can come only after students have practiced and mastered the rules. Like the training wheels used to help small children learn to ride a bicycle, the explanations teachers provide to students learning abstract mathematics are there primarily to maintain the learner’s confidence. Indeed, leaving aside the psychological aspect, there is some evidence that they actually hinder the learning process! A notorious example (within arithmetic) is the common practice among U.S. and British mathematics teachers to introduce multiplication of whole numbers as repeated addition. This practice can cause the students problems throughout the remainder of their mathematical education, and often throughout their adult lives. To cite just one of many studies that have highlighted the problems that result from this approach, see Park & Nunes, 2001. A number of studies in the United States and elsewhere, plus an entire curriculum developed and studied in the Soviet Union by the mathematics educator Vasily Davydov in the second half of the 20th century, suggest that a top-down approach to teaching mathematics is always better in the long run (see Davydov, 1975a, 1975b; Schmittau, 2004). What these studies indicate is that the most efficient way to teach mathematics is to begin by exposing students to issues that will in due course provide examples of a new concept or method, give the students time to explore those examples with guidance, and then teach the new mathematical ideas in a purely abstract, formal, rule-based fashion. Only afterward should teachers show how the issues explored earlier provide examples of what has just been learned formally. This

For most students, algebra is their first encounter with the kind of thinking that is typical of modern mathematics—precise, logical, analytical thinking about pure abstractions. Although experienced mathematicians find such thinking “natural,” that is a result of training and practice. The brain does not find such thinking at all natural. It has to be learned, and the only known way to do so is top-down, by first learning the rules and allowing meaning and understanding to emerge in due course—as it will, because of the way the human brain seeks meaning and understanding in everything it experiences.

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Beyond Algebra When we go from arithmetic to higher mathematics, our understanding of the learning process becomes even less grounded in scientific certainty. To date I am aware of only two attempts to provide scientific theses about the way we learn and do mathematics. Lakoff and Nunez (2000) provide a cognitive science–based theory for the way we learn and understand new mathematics. My own book, The Math Gene (Devlin, 2000), uses the technique of rational reconstruction to provide an evolutionary account of our acquisition of mathematical ability. The Lakoff and Nunez theory is based on the idea of a cognitive metaphor: the individual learning some new mathematics does so by constructing—perhaps unconsciously—an interpretive mapping to the new information from what is already known. Although this thesis is not without problems, at a very basic brain level something like this must be going on. In my theory, I show how mathematical ability is an amalgam of nine basic capacities that our ancestors acquired through natural selection over hundreds of thousands of years: 1. Number sense 2. Numerical ability 3. Spatial-reasoning ability 4. A sense of cause and effect 5. The ability to construct and follow a causal chain of facts or events 6. Algorithmic ability 7. The ability to handle abstraction

of course is a systematic approach to the need to merge bottom-up and top-down thinking, which advances in our understanding of the brain have indicated is required to learn mathematics. (There is some controversy about aspects of these curricula, but the overall need for a combination of top-down and bottom-up approaches that they embody is generally accepted.)

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8. Logical-reasoning ability 9. Relational-reasoning ability

Implications for Teaching Both theories have implications for how mathematics should be taught. However, although some corroborative evidence has been obtained for each theory, further work is required before they can be viewed as in any way definitive. What is of value now is to keep asking the questions that lay behind both of those investigations: • How did the human brain acquire the capacity to do mathematics? • Given a physical brain that is a product of natural selection in the environment of our ancestors, just how can it do mathematics? What makes the first question particularly intriguing is that mathematics is at most a few thousand years old, which is a mere eye-blink on the timescale of evolution. Thus, the capacity for doing mathematics must have been in place long before the brain started to do mathematics. The two questions point to the key issue in learning mathematics: the brain must be able to master new abstractions and to reason at new levels of abstraction. What little evidence there is suggests that this can be done only by a combination of a bottom-up process of reflection about what is already known (meaning giving rise to new rules) and a top-down process of gaining familiarity with formal rules (rules giving rise to new meanings). Different learners seem to fall at different points on a spectrum between these two extremes and so require different forms of teaching.

The brain’s capacity for doing mathematics must have been in place long before it started to do mathematics.

All of these nine capacities were in place by about 75,000 years ago, many much earlier than that. I argue that the key capacity, the ability to handle abstraction, is equivalent to having language.

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References Butterworth, B. (1999). The mathematical brain. Basingstoke, United Kingdom: Macmillan.

Davydov, V. V. (1975b). The psychological characteristics of the “prenumerical” period of mathematics instruction. In L. P. Steffe (Ed.), Children’s capacity for learning mathematics: Soviet studies in the psychology of learning and teaching mathematics (Vol. 7, pp. 109–205). Chicago: University of Chicago Press. Deacon, T. (1997). The symbolic species: The co-evolution of language and the brain. London: Allen Lane. Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press. Devlin, K. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books. Lakoff, G., & Nunez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. Park, J.-H., & Nunes, T. (2001, July–September). The development of the concept of multiplication. Cognitive Development, 16, 763–773. Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19(1), 19–43. Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358, 749–750.

Davydov, V. V. (1975a). Logical and psychological problems of elementary mathematics as an academic subject. In L. P. Steffe (Ed.), Children’s capacity for learning mathematics: Soviet studies in the psychology of learning and teaching mathematics (Vol. 7, pp. 55–107). Chicago: University of Chicago Press.

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